Xinjie Dai

Xinjie Dai, Baiping Zhang, Diancong Jin

In this paper, we investigate the asymptotic distribution of the normalized error for the Mittag--Leffler Euler (MLE) method applied to a class of multidimensional fractional stochastic differential equations. These equations are reformulated as stochastic Volterra equations (SVEs) featuring a non-diagonal, matrix-valued kernel $K(u)=u^{α-1}E_{α,α}(Au^α)$ with singular exponent $α\in (\frac{1}{2}, 1)$. To enhance computational efficiency, the singular kernel is discretized using the left-rectangle rule, posing technical challenges for the theoretical analysis. To address this, we introduce an auxiliary $K$-undiscretized scheme to bridge the gap between the exact solution and the MLE method, integrating Jacod's stable convergence theory for conditional Gaussian martingales with methodologies developed for SVEs. To the best of our knowledge, this is the first work to establish the asymptotic error distribution for numerical methods incorporating non-diagonal matrix-valued kernels.

Xinjie Dai, Xingyu Liu, Diancong Jin et al.

The generalized Langevin equation (GLE) constitutes a fundamental model for describing nonequilibrium dynamics with memory effects. To overcome the numerical challenges arising from superquadratically growing potentials and degenerate noise, we propose and analyze a structure-preserving splitting averaged vector field (AVF) method for a quasi-Markovian GLE. The core advantage of this method lies in its ability to simultaneously preserve the exponential integrability, Malliavin differentiability, and ergodicity of the underlying continuous system. Notably, by leveraging exponential integrability, Malliavin differentiability, and uniform non-degeneracy of the numerical solution, we obtain the existence and smoothness of its probability density function, which converges to that of the exact solution with first-order accuracy. Furthermore, by validating the Lyapunov condition and the minorization condition using a localized technique, we establish the geometric ergodicity of the numerical solution. Finally, numerical experiments are conducted to confirm the theoretical results.